# Inhomogeneous percolation problems and incipient infinite clusters

Published

Journal Article

The authors consider inhomogeneous percolation models with density p c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod - lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model, they prove that in d=2, the borderline value of lambda is lambda b=1/v. If lambda >1/v then, with probability one, there is no infinite cluster, while if lambda <1/v then, with positive probability, the origin is part of an infinite cluster. This result sheds some light on numerical and theoretical predictions of certain properties of incipient infinite clusters. Furthermore, for d>2, the models studied suggest what sort of 'incipient objects' should be examined in random surface models.

### Full Text

### Duke Authors

### Cited Authors

- Chayes, JT; Chayes, L; Durrett, R

### Published Date

- December 1, 1987

### Published In

### Volume / Issue

- 20 / 6

### Start / End Page

- 1521 - 1530

### International Standard Serial Number (ISSN)

- 0305-4470

### Digital Object Identifier (DOI)

- 10.1088/0305-4470/20/6/034

### Citation Source

- Scopus